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This is the first part of the project toward an effective algorithm to evaluate all genus Gromov-Witten invariants of quintic Calabi-Yau threefolds. In this paper, we introduce the notion of Mixed-Spin-P fields, construct their moduli spaces, and construct the virtual cycles of these moduli spaces.

We analyze the torus fixed loci of Mixed Spin P fields moduli, and deduce its localization formulas with explicit factors. An algorithm toward evaluating quintic's Gromov-Witten and Fan-Jarvis-Ruan-Witten invariants is derived.

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In previous work, the notion of Mixed-Spin-P(MSP) fields is introduced and their moduli space is constructed together with a torus action. By applying virtual localization to their virtual classes moduli of MSP fields, polynomial relations among Gromov Witten(GW) and Fan-Jarvis-Ruan-Witten(FJRW) invariants of Fermat quintics are derived. In this paper, we prove a vanishing of a class of torus-fixed loci. This vanishing plays a key role in later proof that in Witten's gauged linear sigma model for Fermat quintics, the FJRW invariants with insertions 2/5 determine the GW invariants of quintic Calabi-Yau threefolds through CY-LG phase transitions.

This is the first part of the project toward proving the BCOV's Feynman graph sum formula of all genera Gromov-Witten invariants of quintic Calabi-Yau threefolds. In this paper, we introduce the notion of N-Mixed-Spin-P fields, construct their moduli spaces, their virtual cycles, their virtual localization formulas, and a vanishing result associated with irregular graphs.

We prove two structure theorems for the Gromov-Witten theory of the quintic threefolds, which together give an effective algorithm for the all genus Gromov-Witten potential functions of quintics. By using these structure theorems, we prove {Yamaguchi-Yau's Polynomial Ring} Conjecture in this paper and prove {Bershadsky-Cecotti-Ooguri-Vafa's Feynman rule} conjecture in the subsequent paper.

We prove the Bershadsky-Cecotti-Ooguri-Vafa's conjecture for all genus Gromov-Witten potentials of the quintic threefolds, by identifying the Feynman graph sum with the $\nmsp$ stable graph sum via an R-matrix action. The Yamaguchi-Yau functional equations and the formulas of $F_1,F_2$, are direct consequences of the BCOV Feynman sum rule.